The image of the stable J-homomorphism
Consider the 2-primary stable J-homomorphism, \[ J: \pi_ i(SO)\to \pi_ i(S^ 0). \] Let \(j>1\) and let \(i+1=8j=(2^ v)(odd)\). Then the image of J is cyclic of order \(2^{v+1}\). This consequence of the solution of the Adams conjecture is proved in this paper by homotopy theory - independent of the Adams Conjecture. A second thrust of this paper is to compute spheres of origin for elements in the 2-primary image of J. For example, with i as above, an element of order \(2^{4a-2}\) in \[ J(\pi_ i(SO))\subset \pi_ i(S^ 0)=\pi_ i(\Omega^{\infty}S^{\infty}) \] is in the image of \(\pi_ i(\Omega^ nS^ n)\) if \(n=8a-2\) (and not if \(n<8a-2)\). The proofs involve bo-resolutions, arguments involving vanishing lines and filtrations in Adams spectral sequences, and the techniques for which the authors are celebrated.
scientific article; zbMATH DE number 4151381